Prandtl Lifting Line Theory

15.6.2.1 Downwash Evolution

A semi-infinite vortex sheet trailing a wing and contained in the (x, y) plane induces at a point in that plane a vertical component w(x, y) called downwash. Near the x-axis (- 2 < y < b), far upstream the effect of the wake decays and the downwash goes to zero. At the lifting line, ww(y) is the downwash induced by a semi-infinite vortex sheet. Far downstream, in the Trefftz plane, the downwash wT (y) results from an infinite vortex sheet. Because of the symmetry, the infinite vortex filaments induce velocities that are twice that of semi-infinite ones. As a result, wT (y) = 2ww(y).

15.6.2.2 Wing with Ideal Loading

As seen in class, the ideal loading of a wing for minimum induced drag is elliptic, i. e.

Theory shows that

r0 = 2UbA1 = 2Ub (Cl’>cruise